Average Error: 6.2 → 2.5
Time: 2.7s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.3249082459376733 \cdot 10^{-158} \lor \neg \left(z \le 6.9390753974297484 \cdot 10^{-149}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.3249082459376733 \cdot 10^{-158} \lor \neg \left(z \le 6.9390753974297484 \cdot 10^{-149}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{z - t}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r278124 = x;
        double r278125 = y;
        double r278126 = z;
        double r278127 = t;
        double r278128 = r278126 - r278127;
        double r278129 = r278125 * r278128;
        double r278130 = a;
        double r278131 = r278129 / r278130;
        double r278132 = r278124 + r278131;
        return r278132;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r278133 = z;
        double r278134 = -5.324908245937673e-158;
        bool r278135 = r278133 <= r278134;
        double r278136 = 6.939075397429748e-149;
        bool r278137 = r278133 <= r278136;
        double r278138 = !r278137;
        bool r278139 = r278135 || r278138;
        double r278140 = y;
        double r278141 = a;
        double r278142 = r278140 / r278141;
        double r278143 = t;
        double r278144 = r278133 - r278143;
        double r278145 = x;
        double r278146 = fma(r278142, r278144, r278145);
        double r278147 = r278144 / r278141;
        double r278148 = r278140 * r278147;
        double r278149 = r278145 + r278148;
        double r278150 = r278139 ? r278146 : r278149;
        return r278150;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.324908245937673e-158 or 6.939075397429748e-149 < z

    1. Initial program 6.9

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]

    2. Simplified2.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]

    if -5.324908245937673e-158 < z < 6.939075397429748e-149

    1. Initial program 4.2

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity4.2

      \[\leadsto x + \frac{y \cdot \left(z - t\right)}{\color{blue}{1 \cdot a}}\]

    4. Applied times-frac3.7

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{z - t}{a}}\]

    5. Simplified3.7

      \[\leadsto x + \color{blue}{y} \cdot \frac{z - t}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.3249082459376733 \cdot 10^{-158} \lor \neg \left(z \le 6.9390753974297484 \cdot 10^{-149}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))